3 Answers
3

Yep, there are various types of spokes, as already explained. There are also various ways of "lacing" the spokes, and various spoke counts.

When picking a spoke lacing scheme, there are six basic considerations:

Strength

Flexibility

Torque resistance

Air resistance

Weight

Appearance/sex/stupidly spending money

Strength is determined by the number of spokes, how they are arranged, and how they interface with the hub and rim. If you look at a normal "three cross" wheel, the spoke leaves the hub at an angle nearly tangential to the hub circumference. This way the stress on the hub is minimized. A "four cross" lacing (commonly seen on heavy touring bikes an the like) goes that one better and causes the spoke to be truly tangential to the hub circumference (plus the increased spoke length gives the wheel more "spring").

You rarely see "two cross" and probably never see "one cross" lacings, but "radial" lacing (where the spokes leave the hub at right angles to the circumference) is becoming fairly common, especially on front hubs of "mountain" or "cross" bikes. Radial lacing places considerable stress on a conventional hub, and with radially laced wheels you'll often see the spokes reversed, with the nipples on the hub end, because the hub must be redesigned to stand the stress. Additionally, both because a radially laced wheel is so stiff and because usually radial wheels have low spoke counts, the stress on the rim is higher, such that the nipples would be apt to pull through. So you'll sometimes see the bent end of the (reversed) spoke engaged in a slot in the rim or some such.

Note that while radial laced wheels are stiffer against radial forces on the wheel, they don't resist torque very well at all -- if a rear wheel were to be radially spoked the torque of the drive train would cause the wheel to twist into a sort of spiral, hindering the transmission of power to the wheel circumference. Similarly, if disk brakes were used on a radial front wheel the wheel would twist severely when the disk brakes were applied.

[However, it should be noted that some people like to lace their rear wheel with a cross pattern on one side and radial spokes on the other. This provides the needed torsional rigidity while permitting half the wheel to be radial, but it's not clear that there's any real advantage to the scheme.]

Air resistance obviously increases with spoke count, and is affected by the profile of the spoke as well. Although air resistance is important to pro racers because the top of the wheel is moving forward at twice the speed of the bicycle, it's probably safe to say that the air resistance due to spokes on a standard 32-spoke three-cross wheel would not be noticed by most average bikers, even at fairly high speeds.

The ultimate low-wind-resistance wheel is the disk wheel, of course, and there are other exotic designs with 1, 2, or 3 broad, flat "spokes" made of high-strength composites bonded to rim and hub. But generally you'll only see the disk wheels on rear wheels because the disk becomes a kite if it gets cross-ways to the wind as would happen on the front wheel.

Weight is reduced by using more exotic materials (especially for the rim), and by reducing spoke count. The more exotic the rim, the more intimately the design of the rim ties into the spoking scheme. Some slight weight reduction is also possible by reducing the "cross" of the spoking (and hence reducing spoke length).

And, of course, sex. Many of the above schemes (especially things like the low-spoke-count radially spoked front wheel with reversed spokes) have little practical benefit for the average cyclist, add cost, and reduce reliability. (Ever heard one of those fancy high-tension spokes go? It sounds like a gunshot. And if one breaks you're likely dead in the water until you can get to a bike shop, whereas with a standard three-cross 32-spoke wheel one can usually "limp home" if a roadside repair is not possible.)

A wheel with many lightweight spokes will be reliable and remain relatively rideable even if one spoke breaks. But all those spokes are like an eggbeater in the wind: not aerodynamic. So wheels like this are mostly used by heavier riders or in touring, where durability trumps speed.

A wheel with fewer spokes puts more load on each of those spokes, and can't tolerate any of them breaking. Also, both to make up for the lower spoke count and because the whole point of reducing spoke count is to improve aerodynamics, these wheels usually have deep-section rims with a teardrop cross-section. These increase the wheel's strength (at the cost of weight), and improve its aerodynamics by replacing what would be the fastest-moving parts of the spokes (at the tips) with solid aerodynamically-shaped rim.

A trispoke or disc wheel is the next logical step. These are much more expensive to fabricate, and discs especially are problematic in crosswinds. They also tend to be heavy, making them inappropriate for some kinds of racing, but not so much triathlons, which are about steady effort.

Bladed spokes are marginally more aerodynamic than round spokes. At the very high end, there are some wheels that use fettucini-like strips of carbon fiber in place of steel spokes—these may cost as much as a very nice bike themselves.

Double-butted spokes are thicker at the ends than in the middle.
The most popular diameters are 2.0/1.8/2.0 mm (also known as 14/15
gauge) and 1.8/1.6/1.8 (15/16 gauge).

Double-butted spokes do more than save weight. The thick ends make
them as strong in the highly-stressed areas as straight-gauge spokes
of the same thickness, but the thinner middle sections make the spokes
effectively more elastic, allowing them to stretch (temporarily) more
than thicker spokes.

As a result, when the wheel is subjected to sharp localized stresses,
the most heavily-stressed spokes can elongate enough to shift some of
the stress to adjoining spokes. This is particularly desirable when
the limiting factor is how much stress the rim can withstand without
cracking around the spoke holes.

Triple-butted spokes, such as the DT Alpine III, are the best
choice when durability and reliability is the primary aim, as with
tandems and bicycles for loaded touring. They share the advantages of
single-butted and double-butted spokes. The DT Alpine III, for
instance, is 2.34 mm (13 gauge) at the head, 1.8 mm (15 gauge) in the
middle, and 2.0 mm (14 gauge) at the threaded end.

Single- and triple-butted spokes solve one of the great problems of
wheel design: Since spokes use rolled, not cut threads, the outside
diameter of the threads is larger than the base diameter of the spoke
wire. Since the holes in the hub flanges must be large enough for the
threads to fit through, the holes, in turn, are larger than the wire
requires. This is undesirable, because a tight match between the spoke
diameter at the elbow and the diameter of the flange hole is crucial
to resisting fatigue-related breakage.

Since single- and triple-butted spokes are thicker at the head end
than at the thread end, they may be used with hubs that have holes
just large enough to pass the thick wire at the head end.

Aero (elliptical) spokes are a variety of double-butted spoke in
which the thin part is swaged into an elliptical cross section, making
these spokes a bit more ærodynamic than round-section spokes. The most
widely available spokes of this type are the Wheelsmith Æro. These are
2.2 x 1.8 mm at the ends, and the middles are equivalent to 16 gauge,
but in the form of a 1.8 x 1.2 mm ellipse. The Wheelsmith Æro is my
favorite spoke for high-performance applications, not just because of
whatever ærodynamic advantage it may offer, but because the flat
center section provides an excellent visual indicator to help the
wheelbuilder eliminate any residual twist in the spoke. This helps
build a wheel that will stay true.

Aero (bladed) spokes have a more pronounced aero shape, flat,
rather than elliptical. Although they are the most aerodynamic of
spokes, they won't normally fit through the holes in a standard hub
because they are too wide. To use "blades", the hub must be slotted
with a file. Slotting the holes can weaken the flange, and will
usually void the warranty of the hub. It is also a lot of trouble.

In theory, "aero" spokes should be more aerodynamic, however if you think about stressing the spoke, it would be more stiff in the direction the wheel is rotating because it is thicker, and have more flex perpendicular to that, since the spoke is thin or flat on the side.

I have double-butted spokes on my cross-country mountain bike, and raced on them for years without untruing the wheels, and it saves a small amount of weight per spoke but its at a point where the spoke isn't heavily stressed, so it retains most of its strength. Especially good for lighter weight riders on full suspension bikes where the wheels won't be subjected to as much of a beating in the first place.